1. Prepared by Dedra Demaree, Georgetown University
Mechanical Waves
Prepared by
Dedra Demaree, Georgetown University
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2. Mechanical Waves How do bats "see" in the dark?
How can you transmit energy without transmitting matter?
Why does the pitch of a train whistle change as it passes you?
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3. Be sure you know how to:
Describe simple harmonic motion verbally, graphically, and algebraically (Section 19.3).
Determine the period and amplitude of vibrations using a graph (Section 19.3).
Determine the power of a process (Section 6.8).
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4. What's new in this chapter
So far we have looked at the vibrational motion of a single object, such as a pendulum.
We have not considered what happens to the environment that is in contact with the vibrating object.
In this chapter we focus on the effect that the vibrating object has on the medium that surrounds it.
An example of such an effect is the propagation of sound.
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5. Observations: Pulses and wave motion
When an object vibrates, it also disturbs the medium surrounding it.
Place a heavy metal cylinder attached to a spring so that the cylinder is partially submerged in a tub with water. Push down on the cylinder a little and release it. The vibrating cylinder (the source) sends ripples (waves) across the tub.
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6. Waves and wave fronts
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7. Wave motion
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8. Two kinds of waves
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9. Reflection of waves
When a wave reaches the wall of the container or the end of the Slinky or rope, it reflects off the end and moves in the opposite direction.
When a wave encounters any boundary between different media, some of the wave is reflected back.
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10. Sound waves If you strike a tuning fork, you hear sound.
You can feel the vibrations of the prongs if you touch them.
If you place the vibrating prongs in water, you see ripples going out.
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11. Mathematical descriptions of a wave
A wave can be created in a rope by a motor that vibrates the end of a rope up and down, producing a transverse wave.
The displacement is described by a sinusoidal function of time:
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12. Mathematical descriptions of a wave
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13. The shape of a transverse wave at five consecutive times
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14. Mathematical description of a traveling sinusoidal wave
We know the source oscillates up and down with a vertical displacement given by:
We can mathematically describe the disturbance y(x, t) of a point of the rope at some positive position x to the right of the source (at x = 0) by:
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15. Wavelength
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16. Tip
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17. Mathematical description of a traveling sinusoidal wave
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18. Tip
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19. Phase
Two points in a medium are "in phase" if at every clock reading, their displacements are exactly the same.
Two points separated by a distance equal to an integer multiple of wavelengths are always in phase.
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20. Tip
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21. Dynamics of wave motion: Speed and the medium
We have defined the speed of a wave as the distance that a disturbance travels in the medium during a time interval divided by that time interval.
This operational definition of speed tells us how to determine the speed but does not explain why a particular wave has a certain speed.
Our goal is to investigate what determines the wave speed.
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22. Observational experiment
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23. Observational experiment
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24. Testing experiment
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25. Wave speed
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26. Tip
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27. Tip
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28. Wave amplitude and energy in a two-dimensional medium
A beach ball bobs up and down in water in simple harmonic motion, producing circular waves that travel outward across the water surface in all directions.
The amplitudes of the crests decrease as the waves move farther from the source.
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29. Wave amplitude and energy in a two-dimensional medium
The circumference of the second ring is two times greater than the first, but the same energy per unit time moves through it.
The energy per unit circumference length passing through the second ring is one-half that passing through the first ring.
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30. Two-dimensional waves produced by a point source
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31. Example 20.3
You do a cannonball jump off a high board into a pool. The wave amplitude is 50 cm at your friend's location, 3.0 m from where you enter the water. What is the wave amplitude at a second friend's location, 5.0 m from where you enter the water?
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32. Wave amplitude and energy in a three-dimensional medium
The area of the second sphere is four times the area of the first sphere, but the same energy per unit time moves through it.
The energy per unit area through the second sphere is one-fourth that through the first sphere.
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33. Three-dimensional waves produced by a point source
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34. Tip
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35. Wave power and wave intensity
The intensity of a wave is defined as the energy per unit area per unit time interval that crosses perpendicular to an area in the medium through which it travels:
The unit of intensity I is equivalent to joules per second per square meter or watts per square meter.
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36. Reflection and impedance
If you hold one end of a rope whose other end is fixed and shake it once, you create a transverse traveling incident pulse.
When the pulse reaches the fixed end, the reflected pulse bounces back in the opposite direction.
The reflected pulse is inverted—oriented downward as opposed to upward.
What happens to a wave when there is an abrupt change from one medium to another?
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37. Reflection and impedance
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38. Impedance
Impedance characterizes the degree to which waves are reflected and transmitted at the boundary between different media.
Impedance is defined as the square root of the product of the elastic and inertial properties of the medium:
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39. Ultrasound
Ultrasound takes advantage of the differing densities of internal structures to "see" inside the body.
The impedance of tissue is much greater than that of air. As a result, most of the ultrasound energy is reflected at the air-body interface and does not travel inside the body.
To overcome this problem, the area of the body to be scanned is covered with a gel that helps "match" the impedance between the emitter and the body surface.
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40. Superposition principle
We have studied the behavior of a single wave traveling in a medium and initiated by a simple harmonic oscillator.
Most periodic or repetitive disturbances of a medium are combinations of two or more waves of the same or different frequencies traveling through the same medium at the same time.
Now we investigate what happens when two or more waves simultaneously pass through a medium.
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41. Observational experiment
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42. Superposition principle
Imagine we have two vibrating sources in water, each of which sends out sinusoidal waves.
Consider point C in the figure.
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43. Superposition principle
Imagine we have two vibrating sources in water, each of which sends out sinusoidal waves.
Consider point D in the figure.
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44. Skills for analyzing wave processes
When problem solving:
Draw displacement-versus-time or displacement-versus-position graphs to represent the waves if necessary.
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45. Superposition principle for waves
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46. Superposition principle for waves
The process in which two or more waves of the same frequency overlap is called interference.
Places where the waves add to create a larger disturbance are called locations of constructive interference.
Places where the waves add to produce a smaller disturbance are called locations of destructive interference.
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47. Huygens' principle
We can explain the formation of the waves using the superposition principle, first explained by Christiaan Huygens (1629–1695).
Each arc represents a wavelet that moves away from a point on the original wave front.
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48. Testing experiment
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49. Sound
We hear sound when pressure variations at the eardrum cause it to vibrate.
Pressure waves in the frequency range that we can hear are called sound.
Higher- or lower-frequency pressure waves are called ultrasound or infrasound.
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50. Loudness and intensity
Loudness is determined primarily by the amplitude of the sound wave: the larger the amplitude, the louder the sound.
Equal-amplitude sound waves of different frequencies will not have the same perceived loudness to humans.
To measure the loudness of a sound, we measure the intensity: the energy per unit area per unit time interval.
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51. intensity level
Because of the wide variation in the range of sound intensities, a quantity called intensity level is commonly used to compare the intensity of one sound to the intensity of a reference sound.
Intensity level is defined on a base 10 logarithmic scale as follows:
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52. Intensities and intensity levels of common sounds
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53. Quantitative Exercise 20.6
The sound in an average classroom has an intensity of 10–7 W/m2.
Determine the intensity level of that sound.
If the sound intensity doubles, what is the new sound intensity level?
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54. Tip
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55. Pitch, frequency, and complex sounds
Pitch is the perception of the frequency of a sound.
Tuning forks of different sizes produce sounds of approximately the same intensity, but we hear each as having a different pitch.
The shorter the length of the tuning fork, the higher the frequency and the higher the pitch.
Like loudness, pitch is not a physical quantity but rather a subjective impression.
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56. Complex sounds, waveforms, and frequency spectra
An oboe and a violin playing concert A equally loudly at 440 Hz sound very different.
Which other property of sound causes this different sensation to our ears?
Musical notes from instruments have a characteristic frequency, but the wave is not sinusoidal.
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57. Waveforms
Superposition causes the sound from two tuning forks to create a complex pattern.
This complex pattern is called a waveform: a combination of two or more waves of different frequencies and possibly different amplitudes.
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58. Frequency spectrum
A spectrum analyzer decomposes a complex waveform into its component frequencies.
The frequency spectrum plots the amplitude of each component versus the frequency.
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59. Fundamental and harmonics
The lowest frequency of a complex wave is called the fundamental.
Higher-frequency components that are whole-number multiples of the fundamental are called harmonics.
We usually identify the pitch as the fundamental frequency.
The complexity of the waveform (the number of harmonics) contributes to the quality of the sound we associate with specific instruments.
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60. Beat and beat frequencies
Two sound sources of similar (but not the same) frequency are equidistant from a microphone that records the air pressure variations due to the two sound sources as a function of time.
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61. Beat and beat frequencies
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62. Standing waves on strings
You shake the end of a rope that is attached to a fixed support.
At specific frequencies, you notice that the rope has large-amplitude sine-shaped vibrations that appear not to be traveling.
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63. Observational experiment
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64. Standing waves on a string
The lowest-frequency vibration of the rope is one up-and-down shake per time interval:
This frequency is called the fundamental frequency.
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65. Testing experiment
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66. Standing waves on a string
Consider the vibration that produces a standing wave on the rope at twice the frequency of the fundamental vibration.
Vibrate the rope so that your hand has two up-and-down shakes during the time interval needed for a pulse to make a round trip.
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67. Standing wave frequencies on a string
We can write this expression in terms of the wavelengths:
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68. Standing wave frequencies on a string (Cont'd)
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69. Standing waves on a string
In a standing wave, only the antinodes reach the maximum displacement.
The nodes do not vibrate at all.
All of the points vibrate in phase.
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70. Example 20.7
A banjo D string is 0.69 m long and has a fundamental frequency of 294 Hz. Shortening the string (by holding the string down over a fret) causes it to vibrate at a higher fundamental frequency. Where should you press to play the note F-sharp, which has a fundamental frequency of 370 Hz?
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71. Standing waves in open-open pipes
A flute is a wind instrument that is an example of an open-open pipe, so called because it is open on both ends.
Blowing into the open-open pipe initiates pressure pulses at that end of it.
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72. Standing wave vibration frequencies in open-open pipes
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73. Standing waves in one-end-closed pipes: An open-closed pipe
Consider a pipe that is closed at one end and open at the other end.
Examples of this type of open-closed pipe include a clarinet, a trumpet, and the human throat.
Blowing into the reed or mouthpiece initiates a high-pressure pulse near the closed end of the pipe.
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74. Standing wave vibration frequencies in open-closed pipes
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75. Exciting the sound and changing the pitch of a wind instrument
In clarinets and saxophones, the sources of vibrations are the reeds. In trumpets, trombones, and French horns, the sources of vibrations are the vibrating lips and mouthpiece.
The vibrations produced by the reeds and mouthpieces are pressure pulses at a variety of frequencies.
The pipes attached to the reeds and mouthpieces reinforce only those input frequency vibrations that are at resonant frequencies of the instruments.
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76. Testing our understanding of standing waves in pipes
If our reasoning is correct that sound in pipe-like instruments is due to standing waves inside of them, then changing the medium (which changes the wave speed) inside a pipe should change the fundamental frequency produced by the pipe, even though the length of the pipe has not changed.
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77. The Doppler effect: Putting it all together
When you hear the sound from a horn of a passing car, its pitch is noticeably higher than normal as it approaches, but noticeably lower than normal as it moves away.
This phenomenon is an example of the Doppler effect.
The Doppler effect occurs when a source of sound and an observer move with respect to each other and/or with respect to the medium in which the sound travels.
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78. Observational experiment
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79. Observational experiment
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80. Doppler effect for the source moving relative to the medium
We use the minus sign when the source is moving toward the observer.
We use the plus sign when the source is moving away from the observer.
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81. Doppler effect for the observer moving relative to the medium
This equation can be used to calculate the frequency fO detected by an observer moving at speed vO toward (plus sign) or away from (minus sign) a stationary source emitting waves at frequency fS.
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82. Doppler effect for sound
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83. Tip
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84. Measuring the speed of blood flow in the body
The Doppler effect is used to measure the speed of blood flow in the body.
Ultrasound waves are directed into an artery. The waves are reflected by red blood cells back to the receiver.
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85. Example 20.9
A friend with a ball attached to a string stands on the floor and swings the ball in a horizontal circle. The ball has a 400-Hz buzzer in it. When the ball moves toward you on one side, you measure a frequency of 412 Hz. When the ball moves away on the other side of the circle, you measure a frequency of 389 Hz. Determine the speed of the ball.
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86. Summary
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87. Summary
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88. Summary
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89. Summary
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